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\section {$g$ function}

The extraction cost function is:
\begin{equation}
g(S,N,s)=\alpha_0s+\alpha_1\ln \biggl\{\frac{\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S}{\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S-s}\biggr \}
\label{eq:TCMining}
\end{equation}

where the terms $\alpha_0$, $\alpha_1$, $\alpha_2$, $\alpha_3$ are parameters. Marginal mining costs are then given by
 \begin{equation}
\frac{\partial g}{\partial s}=\alpha_0+\frac{\alpha_1}{\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S-s} 
 >0 \label{eq:MCMining}
 \end{equation}
We also need other first and second derivatives for the future calculation. The first derivatives of $S$ and $N$:
 \begin{eqnarray}
\frac{\partial g}{\partial S}&=&\frac{\alpha_1s}{(\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S)(\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S-s)} 
 >0 \label{eq:gdS} \\
\frac{\partial g}{\partial N}&=&-\frac{\alpha_1\alpha_2s}{(\alpha_3+N)^2(\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S)(\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S-s)} 
=-\frac{\alpha_2}{(\alpha_3+N)^2}\frac{\partial g}{\partial S}<0
\label{eq:gdN}
\end{eqnarray} 


The second derivatives of function g:
\begin{eqnarray}
\frac{\partial^2g}{\partial s^2} &=& \frac{\partial^2g}{\partial S\partial s} = \frac{\alpha_1}{(\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S-s)^2} >0
\label{eq:gds2}\\
\frac{\partial^2g}{\partial N\partial s} &=& -\frac{\alpha_1\alpha_2}{(\alpha_3+N)^2(\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S-s)^2} = -\frac{\alpha_2}{(\alpha_3+N)^2}\frac{\partial^2g}{\partial s^2}<0 \label{eq:gdNs}\\
\frac{\partial^2g}{\partial N\partial S} &=& -\frac{\alpha_1\alpha_2s(2\bar{S}-\frac{2\alpha_2}{\alpha_3+N}-2S-s)}{[(\alpha_3+N)(\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S)(\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S-s)]^2}  <0\label{eq:gdNS}\\
\frac{\partial^2g}{\partial N^2} &=&
\frac{2\alpha_1\alpha_2s}{(\alpha_3+N)^3(\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S)(\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S-s)} \nonumber\\&+&\frac{\alpha_1\alpha_2^2s(2\bar{S}-\frac{2\alpha_2}{\alpha_3+N}-2S-s)}{[(\alpha_3+N)^2(\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S)(\bar{S}-\frac{\alpha_2}{\alpha_3+N}-S-s)]^2}
>0\label{eq:gdN2}
\end{eqnarray}






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